7th Grade MathDecember 7 – December 11

(Mon/Tues) What are we doing today?

• Area of Circles – We may discuss some problems in Lesson 7 and Lesson 8 but most work is not in the book

• Assignment #5 is online now. Check my website for the link. If you completed the classwork (Nov 30/Dec 1) last week, it will show up as already completed as this is the same assignment. If you turned it in then, I gave you 1 extra credit point as well.

• On Wednesday, I will host an afternoon session for any students who want to get extra review and practice with circumference and area. The link will be sent to your e-mail. I must have at least 2 students in the session to run it.

• Reminder: MATHia is again due on Saturday this week.

• …trying to take over the world

Learning Targets

• I can find the area of a circle given the radius or diameter.

• I can use the formula 𝐴 = 𝜋𝑟2 to find area of a circle.

• I can solve problems related to the area of a circle.

• I understand that squaring a number means multiplying that number by itself. It does not mean multiplying it by 2. For example, 52 = 25but 52 ≠ 10.

• I understand that when discussing area, the units are also squared. For example, square inches, square feet, square miles, etc. are used for measurements of area.

Area (Review)

• Recall that area measures how much space is inside of a figure.

• In the past, you may have seen formulas for the area of a rectangle 𝐴 = 𝑙 ∙ 𝑤 or a

triangle 𝐴 =1

2∙ 𝑏 ∙ ℎ .

• Area can be thought of as “How many squares with a side of 1 unit will fit into this figure?”

Problem #1 (Not in Book)

• On the right, I have measured the radius and area of various circles. The area has been rounded to 2 decimal places.

• Is this relationship proportional? Explain why or why not.

• Can you write a formula for area?

Problem #1 (ANSWER)

• No, this is not proportional.

• In the first row, radius is being multiplied by 3.14 (𝜋). However, in the fourth row, the radius is multiplied by 31.416. In each row, the number being multiplied changes which means it cannot be proportional.

How can we create a formula for area?

• Remember that the distance around a circle, if given the radius, is 2𝜋𝑟.

• If we cut a circle apart and unrolled it, like is seen here, we would have a triangle with a height of r and base of 2𝜋𝑟.

• The formula for area of a triangle is

𝐴 =1

2𝑏ℎ.

• So, we have 𝐴 =1

22𝜋𝑟 ∙ 𝑟 which

simplifies to 𝐴 = 𝜋 ∙ 𝑟 ∙ 𝑟 or, as it is more commonly written 𝐴 = 𝜋𝑟2.

Formula for Area of a Circle

• The formula for area of a circle is 𝐴 = 𝜋𝑟2.

• Notice that 𝑟2 is the not the same thing as 2 ∙ 𝑟. It is the same as 𝑟 ∙𝑟. For example, 72 is equal to 49 and not 14.

• Miscalculating 𝑟2 is one of the most common mistakes when finding the area of a circle.

• If you are given the diameter, you will need to divide by 2 to find the radius to use.

Example #1

• We know the radius is 3 inches. So, we do not have to do anything to get the radius for the area formula.

• 𝐴 = 𝜋𝑟2 → 𝜋 ∙ 32 → 𝜋 ∙ 9 = 9𝜋

• If we want the exact area, it is 9𝜋 in2. If we want an approximate answer, we can multiply 9 by 3.14 to get about 28.26 in2.

• Note: If we have a unit measurement, like inches, the area will be in a square measurement. For example, inches becomes square inches, miles would become square miles, etc.

Problems #1 & #2 (Not in Book)

Problem #1

• What is the area of a circle with a radius of 12 cm?

Problem #2

• What is the area of a circle with a diameter of 15 cm?

Problem #1 (ANSWER)

• What is the area of a circle with a radius of 12 cm?

• Since we know the radius is 12 cm, we do not have to change this value for the formula.

• 𝐴 = 𝜋𝑟2 → 𝐴 = 𝜋 ∙ 122 → 𝐴 = 𝜋 ∙ 144 = 144𝜋

• The exact area is 144𝜋 cm2. If we want an approximation, we can multiply 144 ∙ 3.14 ≈ 452.16. So, this is approximately 452.16 cm2.

• Note #1: If you had the measurement with just cm as the unit, it would not actually be correct as we cannot measure area using linear units.

• Note #2: Notice that I did not specify how to write the answer. Thus, the exact or approximate answer would both be acceptable.

Problem #2 (ANSWER)

• What is the area of a circle with a diameter of 15 cm?

• Since we have the diameter is equal to 15 cm, we need to divide by 2 to get the radius. The radius is 7.5 cm.

• 𝐴 = 𝜋𝑟2 → 𝜋 ∙ 7.52 → 𝜋 ∙ 56.25 = 56.25𝜋

• The area is exactly 56.25𝜋 cm2. If we want an approximation, we can multiply 56.25 ∙ 3.14 ≈ 176.625. Thus, the approximate area is 176.625 cm2.

Problem #3 (Not in Book)

• A circle with a diameter of 38 feet is to be painted on a wall. If each can of paint can cover 350 square feet, how many cans of paint will need to be purchased to complete this job?

Problem #3 (ANSWER)

• A circle with a diameter of 38 feet is to be painted on a wall. If each can of paint can cover 350 square feet, how many cans of paint will need to be purchased to complete this job?

• First, we need to know the area of this circle. Since we have the diameter is 38 ft, we need to divide by 2 to get the radius. The radius is 19 ft.

• 𝐴 = 𝜋𝑟2 → 𝜋 ∙ 192 → 𝜋 ∙ 361 = 361𝜋

• The exact area is 361𝜋. However, we need an estimate to determine the number of paint cans. So, 361 ∙ 𝜋 ≈ 1134.1149 ft2. So, the area of the circle is about 1134.1 square feet.

• To determine the number of cans, we need to divide by 350 as each paint can will cover 350 sq ft. Thus, 1134.1149/350 ≈ 3.2403. It will take approximately 3.24 cans of paint.

• So, the answer is “To paint the circle, we will need 4 cans of paint.” Why round up? Because generally we cannot buy part of a can of paint. So, we need the extra can even if we only use about 0.2403 (less than ¼) of it.

Summary (Area & Circumference)

Practice Problem (Lesson 7)

Practice Problem (ANSWER)

• The circumference is 15𝜋 cm or approximately 47.1 cm.

• The diameter is 15 cm. So, we can just multiply that by 𝜋.

(Wed/Thurs) What are we doing today?

• Unit 3 Lesson 8 & Unit 3 Lesson 9• Reminder: Assignment #5 is

online.• On Wednesday, I will host an

afternoon session for any students who want to get extra review and practice with circumference and area. The link was sent to your e-mail. I must have at least 2 students in the session to run it.

• Reminder: MATHia is again due on Saturday this week.

• …trying to take over the world

Learning Targets

• I can calculate the area of more complicated shapes that include fractions of circles.

• I can write exact answers in terms of 𝜋.

• I can determine, using the context of a problem, whether calculating area or circumference is necessary.

Practice Problem (Lesson 8)

Practice Problem (ANSWER)

• Approximately, the radius would be 12.1 cm, the diameter 24.2 cm, and the area 460 square cm.

• To find the diameter, we can divide the circumference by 𝜋. Thus, 76/3.14 ≈ 24.2.

• The radius would be half of the diameter or 12.1.

• The area is found using the radius: 𝐴 = 𝜋 ∙ 12.12 → 3.14 ∙ 146.41 ≈ 459.72

Activity 9.1

Activity 9.1 (ANSWER)

• Could more than 1 answer be “correct”?

• The diameter of the circle is 800m. So, the radius is 400m.

• 𝐴 = 𝜋𝑟2 → 𝜋 ∙ 4002 = 160000𝜋.

• So, the exact answer is 160000𝜋.

• If you estimate 𝜋 as 3.14, then you get 502,400. If you estimate 𝜋 to be 3.1415, you

get 502,640. If you estimate 𝜋 to be 22

7(which

some books recommend), then you will get 502,857. The most accurate answer is 502,855 (Choice C) which most calculators will find if you use the 𝜋 button.

Estimating Pi

• As you saw on the previous problem, estimating 𝜋 can cause differences in the final answer. In this problem, the difference between the smallest and largest answers was over 400 square meters.

• This is one reason that it is sometimes preferred to leave 𝜋 in the answer. If I had asked to give an exact answer, leaving 𝜋 in the solution, everyone should have gotten 160000𝜋 which would have made things less confusing.

• In some of your future math classes, especially trigonometry, it will be standard to leave 𝜋 in solutions and not estimate it.

• For those who are curious, your first introduction to trigonometry will be in geometry (10th grade). Later classes, including algebra 2 (11th grade) and pre-calculus (12th grade) will expand on your knowledge.

Circumference vs Area

Circumference

• If the problem refers to “length”, “distance”, or “distance around”

• If the problem refers to regular units such as ft, cm, in, mi, etc.

• If the problem involves the distance a wheel travels (as that distance is the circumference for each rotation)

Area

• If the problem involves “covering” or “filling” something

• If the problem refers to square units such as 𝑓𝑡2, 𝑐𝑚2, 𝑖𝑛2, 𝑚𝑖2, etc.

• If the problem involves square tiles or other coverings being used

Circumference or Area (Not in Book)

For each of the following, determine whether it would be better to find the area of a circle or the circumference. Note: You do not have enough information to actually get numerical answers here.

• How much oil would be used to cover a circular pan

• The distance a car travels with 10 rotations of its wheel

• The length of a circular fence

• The amount of carpet used to cover a circular floor

Circumference or Area (ANSWER)

• AREA – the word “cover” is key here, the oil “fills” the pan which is what area measures (the amount inside the figure)

• CIRCUMFERENCE – “distance” and “rotations” are key words here, the distance traveled for 1 rotation is the circumference of the circle

• CIRCUMFERENCE – “length” is the key word, circumference always measures a length which is the perimeter of the circle

• AREA – “cover” is again the key word, carpet would go inside the edge of the circle which makes it an area measurement

• How much oil would be used to cover a circular pan

• The distance a car travels with 10 rotations of its wheel

• The length of a circular fence

• The amount of carpet used to cover a circular floor

Problem (Not in Book)

• What is the perimeter of this figure? The figure is created by 3 squares and 2 semi-circles.

• Hint: The interior lines (2 of them) do not count as they are not on the perimeter of this figure.

Problem (ANSWER)

• The area is 40 + 20𝜋 or, if written in the other order, 20𝜋 + 40. This is approximately 102.83.

• First, we can find the circumference of the circle. The diameter is 20 (two squares). So, the circumference is 20𝜋. Each semi-circle is half of this, but we need both. So, we can keep that value without dividing it.

• Then, we need the straight sides of the squares. Each is 10. We have four of them (in red here) on the perimeter. So, we have 40.

• We can add 20𝜋 + 40 to find the answer now.

• Note: Your answer could have been exact or approximate.

Practice Problem (Lesson 9)

Practice Problem (ANSWER)

• The area is 540 − 65.25𝜋. Since it says to “express your answer in terms of 𝜋” you should leave it in this form.

• To find this, we first need the area of the entire rectangle. The width is 18. The height is 30 (if we add all the measurements). So, we multiply to get 540.

• Next, we need to find the area of each circle. The radii are half of the measurements given. So, we have radii of 3, 4.5, and 6. We can square each of these to get 9, 20.25, and 36. Each of these are multiplied by 𝜋 to get the area of the circle. So, we have 9𝜋, 20.25𝜋, and 36𝜋. We have to add these together to get 65.25𝜋. This is the part “cut out” of the rectangle.

• Finally, we subtract the cut-out parts from the whole rectangle. So, we have 540 (rectangle) minus 65.25𝜋(circles).

(Fri) What are we doing today?

• Finish Previous Lesson(s) – We will look at 1 or 2 problems we did not get to finish this week. The problem(s) will do will vary by class period but are included in the slides for Wed/Thurs.

• Reminder: Assignment #5 is online.

• UNIT 2 EXAM (REDO) is ONLINE. It is due at 4:00pm on December 18.

• Unit 3 Exam will be next Friday.

• Reminder: MATHia is again due on Saturday this week. No MATHia is due next week.

• …trying to take over the world